Understand the figure/text

The cyclist moves on a circular ground of radius $R=2\text{ km}.$

He starts at point $P$ and reaches point $S$ along the circumference.

The displacement depends only on initial and final positions, so it is the straight-line distance $PS$.

Identify the relative positions of $P$ and $S$

In the standard circular-ground diagram for such questions, points $P, Q, R, S$ divide the circle into four equal parts, so $P$ and $S$ are adjacent quarter points. Hence the angle subtended at the center is $\angle POS = 90^\circ.$

Find the chord length $PS$

The displacement is the chord corresponding to $90^\circ$: $PS = 2R\sin\frac{\theta}{2} = 2(2)\sin 45^\circ.$

Since $\sin 45^\circ = \frac{1}{\sqrt2},$ we get $PS = 4\cdot \frac{1}{\sqrt2} = \frac{4}{\sqrt2} = 2\sqrt2 = \sqrt8\text{ km}.$

Match with options

$\sqrt8\text{ km}$ corresponds to Option A.

Comparison with stored answer

Stored correct answer: A
Derived answer: A

They match.